Rotorcraft Tail Rotor Sizing

Tail Rotor Sizing (flight-mechanics/performance/rotorcraft-tail-rotor-sizing)

Use when you must size the anti-torque tail rotor of a single-main-rotor rotorcraft from the main rotor torque balance: the main rotor power at the rotor speed sets the shaft torque, the anti-torque thrust follows from the torque over the tail arm, and a maximum disk loading sets the tail rotor disk area and radius. The induced velocity and ideal power come from momentum theory, and the total power adds the induced-power factor and a tail-rotor profile power estimate. The main rotor power is an INPUT to this leaf: it is never computed from weight and geometry here. Pure Python, stdlib only. It pairs with flight-mechanics/performance/rotorcraft-hover-performance, which owns the main rotor hover power, and with the forward flight, climb and ground effect rotorcraft leaves for the other flight states.

Domain quick reference

• Main rotor shaft torque: Q = P / omega, from the main rotor power P and the rotor angular speed omega.
• Anti-torque thrust: T_tr = margin_factor * Q / l_arm, with l_arm the tail arm and margin_factor the yaw control margin on the required thrust (default 1.0).
• Tail rotor disk area at the ceiling: A = T_tr / DL_max, with DL_max the maximum disk loading (default 300 Pa).
• Tail rotor radius: R = sqrt(A / PI).
• Disk loading: DL = T_tr / A, at or below DL_max when sized with it.
• Ideal induced velocity (momentum theory): v_i = sqrt(T_tr / (2 * rho * A)), rho = 1.225 kg/m3 at sea level.
• Ideal power: P_ideal = T_tr * v_i.
• Profile power estimate: P_profile = (1/8) * rho * sigma * Cd * A * V_tip^3, with solidity sigma = 0.10, Cd = 0.012 and tip speed V_tip = 200 m/s defaults.
• Total power: P_total = k * P_ideal + P_profile, with induced-power factor k = 1.15.
• SI units throughout: W, rad/s, N m, N, m, m2, Pa, m/s.
• FAR-29 frames transport rotorcraft certification context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the operating point: main rotor power input P, rotor speed omega and tail arm l_arm (main_rotor_torque, tail_rotor_thrust).
2. Choose the yaw margin factor on the required thrust (default 1.0) and the maximum disk loading ceiling.
3. Size the disk: tail_rotor_area at the ceiling, then tail_rotor_radius and confirm the achieved tail_rotor_disk_loading.
4. Compute the momentum theory induced velocity with tail_rotor_induced_velocity and the ideal power with tail_rotor_ideal_power.
5. Estimate the profile power with tail_rotor_profile_power (solidity, Cd, tip speed defaults) and combine with tail_rotor_total_power.
6. For the full chain in one call, run tail_rotor_sizing with all inputs and read the nine documented outputs.
7. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_tail_rotor_sizing.py.

Worked example

Main rotor power 400 000 W at 27 rad/s, tail arm 8.0 m, max disk loading 300 Pa, rho = 1.225 kg/m3, margin factor 1.0, solidity 0.10, Cd 0.012, tip speed 200 m/s, k = 1.15. Module outputs:

• Main rotor torque: Q = 400000 / 27 = 14 814.8 N m.
• Anti-torque thrust: T_tr = 14814.8 / 8.0 = 1851.9 N.
• Disk area: A = 1851.9 / 300 = 6.1728 m2.
• Tail rotor radius: R = sqrt(6.1728 / PI) = 1.4017 m.
• Disk loading: 300.0 Pa, exactly the ceiling.
• Induced velocity: v_i = sqrt(1851.9 / (2 * 1.225 * 6.1728)) = 11.066 m/s.
• Ideal power: P_ideal = 1851.9 * 11.066 = 20 492.0 W.
• Profile power: P_profile = (1/8) * 1.225 * 0.10 * 0.012 * 6.1728 * 200^3 = 9074.1 W.
• Total power: P_total = 1.15 * 20492.0 + 9074.1 = 32 639.8 W.

All values sit inside the spec magnitude bounds (torque 13 000-17 000 N m, thrust 1500-2200 N, area 5.0-7.5 m2, radius 1.2-1.6 m, induced velocity 9-13 m/s, ideal power 15 000-26 000 W, total power 22 000-36 000 W).

Verification

• Confirm tail_rotor_sizing(400000.0, 27.0, 8.0) returns the worked example values above at the default ceiling, rho and factor settings.
• Confirm the torque balance identity: tail_rotor_thrust(Q, arm) * arm equals Q at margin factor 1.0.
• Confirm the radius round-trip: PI * tail_rotor_radius(A)**2 recovers A.
• Confirm the disk loading never exceeds the ceiling when the disk is sized with that ceiling, even with a margin factor above 1.0.
• Confirm the total power identity P_total = k * P_ideal + P_profile.
• Confirm every non-positive power, torque, arm, disk loading, rho, solidity, drag coefficient, tip speed, area and induced-power factor raises ValueError.
• No RNG anywhere: repeated runs give identical floats.
• Run the contract test offline: python3 scripts/test_rotorcraft_tail_rotor_sizing.py (47 tests, deterministic).

Pitfalls

• Feeding a main rotor power computed from weight and geometry: the main rotor power is an input to this leaf and is never derived from the rotorcraft mass; recompute it with the hover or forward-flight sibling for the flight state in question.
• Reading the disk loading ceiling as achieved: A = T_tr/DL_max sizes the disk at the ceiling, so the achieved DL equals DL_max at margin factor 1.0 and stays at or below it when the margin factor is raised.
• Forgetting the torque identity: anti-torque thrust times the tail arm must equal the main rotor shaft torque at margin factor 1.0 (tail_rotor_thrust(Q, arm) * arm == Q); a mismatch flags inconsistent inputs.
• Changing the yaw margin silently: margin_factor scales the required anti-torque thrust, so the disk area and radius grow with it; the defaults for profile power (solidity 0.10, Cd 0.012, V_tip 200 m/s) and k = 1.15 are documented assumptions, not outputs.
• Non-positive power, torque, arm, disk loading, rho, solidity, drag coefficient, tip speed, area or induced-power factor raise ValueError; runs are deterministic (no RNG).

Related leaves

• flight-mechanics/performance/rotorcraft-hover-performance: the main rotor OGE hover power that is the torque-balance input for this leaf.
• flight-mechanics/performance/rotorcraft-forward-flight-performance: main rotor power in forward flight, the other power state feeding the torque balance.
• flight-mechanics/performance/rotorcraft-vertical-climb-performance: climb power states for the same main rotor.
• flight-mechanics/performance/rotorcraft-hover-ground-effect: ground effect on the main rotor power input near the ground.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_rotorcraft_tail_rotor_sizing.py

The test covers the worked example anchors at 4 significant figures and the spec magnitude bounds, every function against its defining equation, the torque-balance round trip, the area-radius round trip, disk loading at or below the ceiling with a yaw margin, the exact key set of the convenience dict, run-to-run determinism, and ValueError rejection of every non-physical input class.

Compliance

• Standards referenced, not reproduced: FAR-29 is a regulatory standard; the tail rotor sizing relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.